I 520 ] 



dicularly on thefe circles, the remainder of the fphere =4x|''3 = 

 1 cube on the diameter of the fphere, as M. Boffut has announced 

 in the fecond volume of the French National Inflitute*. 



2. If a — . fm being a numerical coefficient) then the 



<corred fluents of 



' ar aX r^ — zAi = | ;« x r* — r^ — z^ ]^ 



r 



when z — r the furface becomes ^ m r^ and the folid IjVi r'>> . 

 The curve in this cafe fatisfying the problem cannot pafs through 



Fig. 2. B, for by taking the fluents a = , and therefore if the 



curve pafled through B, m = 2 a = femi-circumference of a circle, 

 the rad. of which is unity, and not an afllgnable number, con- 

 fequently the folid and furface could not be algebraically ex- 

 prefled . 



If 



*By help of the theorem of Archimedes for the furface of a fpherical zone Vivianl's 

 conclufion may, without difficulty, be accurately deduced by method known to geome- 

 tricians before the invention of fluxions and by the help of the lemma above given 

 Boflut's theorem might alfo be deduced by the fame method, independent of fluxions. 

 Hence Viviani's aflertion refpedling the difficulty of folving the problem from geome- 

 trical confideration alone, cannot be confidered as juft. 



